Andrew Wiles' analytic proof of Fermat's Last Theorem FLT, which appeals to geometrical properties of real and complex numbers, leaves two questions unanswered: (i) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? (ii) Why is x^n+y^n=z^n solvable only for n<3? In this inter-disciplinary perspective, we offer insight into, and answers to, both queries; yielding a pre-formal proof of why FLT can be treated as a true (...) arithmetical proposition (one which, moreover, might not be provable formally in the first-order Peano Arithmetic PA), where we admit only elementary (i.e., number-theoretic) reasoning, without appeal to analytic properties of real and complex numbers. We cogently argue, further, that any formal proof of FLT needs---as is implicitly suggested by Wiles' proof---to appeal essentially to formal geometrical properties of formal arithmetical propositions. (shrink)

Computational complexity has often been ignored in the philosophy of mind, in philosophical artificial intelligence studies. The purpose of this paper is threefold. First and foremost, to show the importance of complexity rather than computability in philosophical and AI problems. Second, to rephrase the notion of computability in terms of solvability, i.e., treating computability as non-sufficient for establishing intelligence. The Church-Turing thesis is therefore revisited and rephrased in order to capture the ontological background of spatial and temporal complexity. Third, (...) to emphasize ontological differences between different time complexities, which seem to provide a solid base for a better understanding of artificial intelligence in general. (shrink)

This article develops an axiom system to justify an additive representation for a preference relation ${\succsim}$ on the product ${\prod_{i=1}^{n}A_{i}}$ of extensive structures. The axiom system is basically similar to the n-component (n ≥ 3) additive conjoint structure, but the independence axiom is weakened in the system. That is, the axiom exclusively requires the independence of the order for each of single factors from fixed levels of the other factors. The introduction of a concatenation operation on each factor A i (...) makes it possible to yield a special type of restricted solvability, i.e., additive solvability and the usual cancellation on ${\prod_{i=1}^{n}A_{i}}$ . In addition, the assumption of continuity and completeness for A i implies a stronger type of solvability on A i . The additive solvability, cancellation, and stronger solvability axioms allow the weakened independence to be effective enough in constructing the additive representation. (shrink)

Guessing the outcome of iterations of even most simple arithmetical functions could be an extremely hazardous experience. Not less harder, if at all possible, might be to prove the veracity of even a "sure" guess concerning iterations : this is the case of the famous 3x+1 conjecture. Our purpose here is to study and conceptualize some intuitive insights related to the ultimate (un)solvability of this conjecture.

Climate change mitigation has become a paradigm case both for externalities in general and for the game-theoretic model of the Tragedy of the Commons (ToC) in particular. This situation is worrying, as we have reasons to suspect that some models in the social sciences are apt to be performative to the extent that they can become self-fulfilling prophecies. Framing climate change mitigation as a hardly solvable coordination problem may force us into a worse situation, by changing real-world behaviour to fit (...) our model, rather than the other way around. But while this problem of the performativity of the ToC has been noted in a recent paper in this journal by Matthew Kopec, his proposed strategies for dealing with their self-fulfilling nature fall short of providing an adequate solution. Instead of relying on the idea that modelling assumptions are always strictly speaking false, this paper shows that the problem may be better framed as a problem of underdetermination between competing explanations. Our goal here is to provide a framework for choosing between this set of competing models that allows us to avoid a ‘Russian Roulette’-like situation in which we gamble with existential risk. (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 (classical) 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)

We overview logical and computational explanations of the notion of tractability as applied in cognitive science. We start by introducing the basics of mathematical theories of complexity: computability theory, computational complexity theory, and descriptive complexity theory. Computational philosophy of mind often identifies mental algorithms with computable functions. However, with the development of programming practice it has become apparent that for some computable problems finding effective algorithms is hardly possible. Some problems need too much computational resource, e.g., time or memory, to (...) be practically computable. Computational complexity theory is concerned with the amount of resources required for the execution of algorithms and, hence, the inherent difficulty of computational problems. An important goal of computational complexity theory is to categorize computational problems via complexity classes, and in particular, to identify efficiently solvable problems and draw a line between tractability and intractability. -/- We survey how complexity can be used to study computational plausibility of cognitive theories. We especially emphasize methodological and mathematical assumptions behind applying complexity theory in cognitive science. We pay special attention to the examples of applying logical and computational complexity toolbox in different domains of cognitive science. We focus mostly on theoretical and experimental research in psycholinguistics and social cognition. (shrink)

State-dependent utility is a problem for the behavioural branch of decision theory under uncertainty. It questions the very possibility that beliefs be revealed by choice data. According to the current literature, all models of beliefs are equally exposed to the problem. Moreover, the problem is solvable only when the decision-maker can influence the resolution of uncertainty. This article gives grounds to reject these two views. The various models of beliefs can be shown to be unequally exposed to the problem of (...) state-dependent utility. The problem can be argued to be solved even when the decision-maker has no influence over the resolution of uncertainty. The implications of such reappraisal for a philosophical appreciation of the revealed preference methodology are discussed. (shrink)

Scientists working within a paradigm must play by the rules of the game of that paradigm in solving problems, and that is why incommensurability arises when the rules of the game change. If we deny the thesis of the priority of paradigms, then there is no good argument for the incommensurability of theories and thus for taxonomic incommensurability, because there is no invariant way to determine the set of results provable, puzzles solvable, and propositions cogently formulable under a given paradigm.

Philosophical food ethics or deliberative inquiry into the moral norms for production, distribution and consumption of food is contrasted with food ethics as an international social movement aimed at reforming the global food system. The latter yields an activist orientation that can become embroiled in self-defeating impotency when the complexity and internal contradictions of the food system are more fully appreciated. However, recent work in intersectionality offers resources that are useful to both philosophical and activist food ethics. For activists, intersectionality (...) provides a way to preserve and strengthen the meaningfulness of protest and resistance, even in the face of complexity and uncertain outcomes. For philosophers, intersectionality chastens the tendency to regard moral problems as inherently solvable, and provides a way use tensions inherent in food system reform as a source of ethical insight. (shrink)

This chapter discusses the fundamental question of the defense industry’s role and legitimacy for societies. It begins with a parable of a psychopath doing something self-serving that has beneficial moral consequences. Analogously, it is argued, the defense industry profiting by selling weapons that can kill people makes it useful in solving moral problems not solvable by people with ordinary moral scruples. Next, the chapter argues that while the defense industry is a business, it is also implicated in the security of (...) its home nations. In addition, its products can be used for good or evil. Thus, it must navigate conflicting values, with the result that the rules often thought to rightly regulate the industry—codes of business ethics, engineering ethics, procurement ethics, plus laws of the land, and dicta of conscience—need supplementing by general moral reflection. This chapter argues for a principle that tells how to decide when to violate the rules, for a theory of rational choice on which choosing by this principle is rational, and for teaching that theory to defense industry employees to ensure their ethical behavior. (shrink)

Two plausible claims seem to be inconsistent with each other. One is the idea that if one reasonably believes that one ought to fi, then indeed, on pain of acting irrationally, one ought to fi. The other is the view that we are fallible with respect to our beliefs about what we ought to do. Ewing’s Problem is how to react to this apparent inconsistency. I reject two easy ways out. One is Ewing’s own solution to his problem, which is (...) to introduce two different notions of ought. The other is the view that Ewing’s Problem rests on a simple confusion regarding the scope of the ought-operator. Then, I discuss two hard ways out, which I label objectivism and subjectivism, and for which G.E. Moore and Bishop Butler are introduced as historical witnesses. These are hard ways out because both of these views have strong counterintuitive consequences. After explaining why Ewing’s Problem is so difficult, I show that there is conceptual room in-between Moore and Butler, but I remain sceptical whether Ewing’s Problem is solvable within a realist framework of normative facts. (shrink)

The problem of algorithmic structuring of proofs in the sequent calculi LK and LKB ( LK where blocks of quantifiers can be introduced in one step) is investigated, where a distinction is made between linear proofs and proofs in tree form. In this framework, structuring coincides with the introduction of cuts into a proof. The algorithmic solvability of this problem can be reduced to the question of k-l-compressibility: "Given a proof of length k , and l ≤ k : (...) Is there is a proof of length ≤ l ?" When restricted to proofs with universal or existential cuts, this problem is shown to be (1) undecidable for linear or tree-like LK-proofs (corresponds to the undecidability of second order unification), (2) undecidable for linear LKB-proofs (corresponds to the undecidability of semi-unification), and (3) decidable for tree-like LKB -proofs (corresponds to a decidable subprob- lem of semi-unification). (shrink)

Suppose you want to do as much good as possible. What should you do? According to members of the effective altruism movement—which has produced much of the thinking on this issue and counts several moral philosophers as its key protagonists—we should prioritise among the world’s problems by assessing their scale, solvability, and neglectedness. Once we’ve done this, the three top priorities, not necessarily in this order, are (1) aiding the world’s poorest people by providing life-saving medical treatments or alleviating (...) poverty itself, (2) preventing global catastrophic risks, such as those posed by nuclear war or rogue artificial intelligence, and (3) ending factory farming. These claims are both plausible and striking. If correct, they should prompt a stark revision of how we approach our altruistic activities. However, the project of determining how to do the most good—as opposed to say, whether we should do the most good—has only recently, within the last ten years, become the subject of serious academic attention. Many key claims have not yet been carefully scrutinised. This is a cause for concern: are effective altruists doing good badly? In this thesis, I critique and develop some of the latest claims about how individuals can do the most good. I do this in three areas: the value of saving lives (preventing premature deaths), how best to improve lives (making people happier during their lives), and cause prioritisation methodology (frameworks for determining which problems are the highest priorities). In each case, I raise novel theoretical considerations that, when incorporated, change the analysis. Roughly speaking, my main conclusions are (1) saving lives is not as straightforwardly good we tend to suppose, may not be good at all, and is not clearly a priority; (2) happiness can be measured through self-reports and, based on the self-reported evidence, treating mental health stands out as an overlooked problem that may be an even more cost-effective way to improve lives than alleviating poverty; (3) the cause prioritisation methodology proposed by effective altruists needs to be moderately reconceptualised and, when it is, it turns out it is not as illuminating a tool as we might have thought and hoped. (shrink)

Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two (...) questions unanswered: (i) Why is x^n +y^n = z^n solvable only for n < 3 if x, y, z, n are natural numbers? (ii) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? Prevailing post-Wiles wisdom---leaving (i) essentially unaddressed---dismisses Fermat's claim as a conjecture without a plausible proof of FLT. -/- However, we posit that providing evidence-based answers to both queries is necessary not only for treating FLT as significant, but also for understanding why FLT can be treated as a true arithmetical proposition. We thus argue that proving a theorem formally from explicit, and implicit, premises/axioms using rules of deduction---as currently accepted---is a meaningless game, of little scientific value, in the absence of evidence that has already established---unambiguously---why the premises/axioms and rules of deduction can be treated, and categorically communicated, as pre-formal truths in Marcus Pantsar's sense. Consequently, only evidence-based, pre-formal, truth can entail formal provability; and the formal proof of any significant mathematical theorem cannot entail its pre-formal truth as evidence-based. It can only identify the explicit/implicit premises that have been used to evidence the, already established, pre-formal truth of a mathematical proposition. Hence visualising and understanding the evidence-based, pre-formal, truth of a mathematical proposition is the only raison d'etre for subsequently seeking a formal proof of the proposition within a formal mathematical language (whether first-order or second order set theory, arithmetic, geometry, etc.) By this yardstick Andrew Wiles' proof of FLT fails to meet the required, evidence-based, criteria for entailing a true arithmetical proposition. -/- Moreover, we offer two scenarios as to why/how Fermat could have laconically concluded in his recorded marginal noting that FLT is a true arithmetical proposition---even though he either did not (or could not to his own satisfaction) succeed in cogently evidencing, and recording, why FLT can be treated as an evidence-based, pre-formal, arithmetical truth (presumably without appeal to properties of real and complex numbers). It is primarily such a putative, unrecorded, evidence-based reasoning underlying Fermat's laconic assertion which this investigation seeks to reconstruct; and to justify by appeal to a plausible resolution of some philosophical ambiguities concerning the relation between evidence-based, pre-formal, truth and formal provability. (shrink)

Truths will be defined as an agreement on uncertainties, the consensus over matters of empirical and social nature such as mathematics, physics or economics. As illustrated by Dennis Lindley , ‘individuals tend to know things to be true and false but the extent of this truth and falsity would always remain unknown’. Leading individuals to a permanent state of stress, uncertainty becomes a risk for the social community. Problems could not be presumed to be solvable as any kind of solution (...) would be perceived as uncertain. The struggle for solutions would develop into a state of insecurity and conflict where no individual could trust the answers given by any other member of the community. To prevent a ‘war of all against all’ that would cause anxiety and permanent conflict, it will become an essential duty for the social community to diminish uncertainty by imposing truths. With those, individuals can be reassured about the certainty of their knowledge and use truths as a resolution tool for any dispute that might occur within the social space. For the purpose of social peace, truths will therefore be every outcome of the process through which uncertainty is transformed into an agreed form of knowledge. “Gravity”, “Death” and “God” will all be considered truths as they represent a form of agreement over different domains of the human and social life. It is not the intention of this essay to examine each of the truths separately but rather, to analyse them all together in order to formulate a hypothesis for all empirical and social matters. (shrink)

This chapter focuses on technological innovation and how insights from technological design can be used to address the challenges associated with the setting in which frugal innovation operates. The resource-constrained setting of frugal innovation puts high demands the design requirements of frugal innovation technologies and the possible conflicts between these requirements. Within the ethics of technology, there is a growing literature that explicitly focuses on how to make technological design more sensitive to important moral values, commonly referred to as value-sensitive (...) design or design for values. However, despite strong commonalities, frugal innovation does not feature as a strong application domain in the literature on value-sensitive design practices. Since frugal innovation takes place in and/or for a resource-constrained context, focusing on just one value could easily lead to other relevant values not being appropriately embedded in the design. For value-sensitive design practices to contribute to frugal innovation, it seems better to think in terms of ‘design for context X’ rather than ‘design for value X’, as the latter may be too narrowly focused on one specific value. Systematic research on design practices is necessary to gain more insight in which values are particularly relevant, both in terms of internal values and in terms of external values, but also in the relevant operationalisations, which may differ substantially in different contexts and which may make value conflicts both more complicated but also more easily solvable. (shrink)

The problem addressed in this article is the relationship between law and morality. It is asked (1) to what extent law and morality are connected and separated and (2) since when has it been so. To the extent that law and morality are distinct normative orders, it is asked (3) whether they rule exactly the same behaviors or whether each order rules dierent kinds of behaviors. If they rule at least some of the same behaviors, it is asked (4) whether (...) there can be antinomies (contradictions) between them. If there are an- tinomies, it is asked (5) whether the antinomies are only apparent (prima facie) and are therefore mistakes of human reason, or are definite and real. If the antinomies are apparent or real, it is asked (6) whether law or morality prevails (or should prevail) in the case of an antinomy. If one of these prevails, it is asked (7) whether this is always so, or whether law sometimes prevails (and should prevail) over morality and vice versa. In the case of existing coherence or at least solvable antinomies between law and morality, it is asked (8) whether the consequent achieved unity of practical reason is a specifically moral unity and whether it is a matter of cognition, of institutionalization, of individual or collective construction, or of consensus. (shrink)

The title of the present book suggests that scientific results obtained in mathematics and quantum physics can be in some way related to the question of the existence of God. This seems possible to us, because it is our conviction that reality in all its dimensions is intelligible. The really impressive progress in science and technology demonstrates that we can trust our intellect, and that nature is not offering us a collection of meaningless absurdities. We first of all intend to (...) show with results taken from mathematics and quantum physics: Mathematical Undecidability: man will never have a universal method to solve any mathematical problem. In arithmetic there always will be unsolved, solvable problems. Quantum Nonlocality: certain phenomena in nature seem to imply the existence of correlations based on faster-than-light influences. These influences, however, are not accessible to manipulation by man for use in, for example, faster than light communication. In the various contributions pieces of a puzzle are offered, which suggest that there exists more than the world of phenomena around us. The results discussed point to intelligent and unobservable causes governing the world. One is led to perceive the shade of a reality which many people would call God. (shrink)