A fundamental problem in artificial intelligence is that nobody really knows what intelligence is. The problem is especially acute when we need to consider artificial systems which are significantly different to humans. In this paper we approach this problem in the following way: we take a number of well known informal definitions of human intelligence that have been given by experts, and extract their essential features. These are then mathematically formalised to produce a general measure of intelligence for arbitrary machines. (...) We believe that this equation formally captures the concept of machine intelligence in the broadest reasonable sense. We then show how this formal definition is related to the theory of universal optimal learning agents. Finally, we survey the many other tests and definitions of intelligence that have been proposed for machines. (shrink)

Can humans get arbitrarily capable reinforcement learning agents to do their bidding? Or will sufficiently capable RL agents always find ways to bypass their intended objectives by shortcutting their reward signal? This question impacts how far RL can be scaled, and whether alternative paradigms must be developed in order to build safe artificial general intelligence. In this paper, we study when an RL agent has an instrumental goal to tamper with its reward process, and describe design principles that prevent instrumental (...) goals for two different types of reward tampering. Combined, the design principles can prevent reward tampering from being an instrumental goal. The analysis benefits from causal influence diagrams to provide intuitive yet precise formalizations. (shrink)

Can an agent's intelligence level be negative? We extend the Legg-Hutter agent-environment framework to include punishments and argue for an affirmative answer to that question. We show that if the background encodings and Universal Turing Machine (UTM) admit certain Kolmogorov complexity symmetries, then the resulting Legg-Hutter intelligence measure is symmetric about the origin. In particular, this implies reward-ignoring agents have Legg-Hutter intelligence 0 according to such UTMs.

The first decade of this century has seen the nascency of the first mathematical theory of general artificial intelligence. This theory of Universal Artificial Intelligence (UAI) has made significant contributions to many theoretical, philosophical, and practical AI questions. In a series of papers culminating in book (Hutter, 2005), an exciting sound and complete mathematical model for a super intelligent agent (AIXI) has been developed and rigorously analyzed. While nowadays most AI researchers avoid discussing intelligence, the award-winning PhD thesis (Legg, (...) 2008) provided the philosophical embedding and investigated the UAI-based universal measure of rational intelligence, which is formal, objective and non-anthropocentric. Recently, effective approximations of AIXI have been derived and experimentally investigated in JAIR paper (Veness et al. 2011). This practical breakthrough has resulted in some impressive applications, finally muting earlier critique that UAI is only a theory. For the first time, without providing any domain knowledge, the same agent is able to self-adapt to a diverse range of interactive environments. For instance, AIXI is able to learn from scratch to play TicTacToe, Pacman, Kuhn Poker, and other games by trial and error, without even providing the rules of the games. These achievements give new hope that the grand goal of Artificial General Intelligence is not elusive. This article provides an informal overview of UAI in context. It attempts to gently introduce a very theoretical, formal, and mathematical subject, and discusses philosophical and technical ingredients, traits of intelligence, some social questions, and the past and future of UAI. (shrink)

Increasingly encompassing models have been suggested for our world. Theories range from generally accepted to increasingly speculative to apparently bogus. The progression of theories from ego- to geo- to helio-centric models to universe and multiverse theories and beyond was accompanied by a dramatic increase in the sizes of the postulated worlds, with humans being expelled from their center to ever more remote and random locations. Rather than leading to a true theory of everything, this trend faces a turning point after (...) which the predictive power of such theories decreases (actually to zero). Incorporating the location and other capacities of the observer into such theories avoids this problem and allows to distinguish meaningful from predictively meaningless theories. This also leads to a truly complete theory of everything consisting of a (conventional objective) theory of everything plus a (novel subjective) observer process. The observer localization is neither based on the controversial anthropic principle, nor has it anything to do with the quantum-mechanical observation process. The suggested principle is extended to more practical (partial, approximate, probabilistic, parametric) world models (rather than theories of everything). Finally, I provide a justification of Ockham's razor, and criticize the anthropic principle, the doomsday argument, the no free lunch theorem, and the falsifiability dogma.", . (shrink)

Understanding inductive reasoning is a problem that has engaged mankind for thousands of years. This problem is relevant to a wide range of fields and is integral to the philosophy of science. It has been tackled by many great minds ranging from philosophers to scientists to mathematicians, and more recently computer scientists. In this article we argue the case for Solomonoff Induction, a formal inductive framework which combines algorithmic information theory with the Bayesian framework. Although it achieves excellent theoretical results (...) and is based on solid philosophical foundations, the requisite technical knowledge necessary for understanding this framework has caused it to remain largely unknown and unappreciated in the wider scientific community. The main contribution of this article is to convey Solomonoff induction and its related concepts in a generally accessible form with the aim of bridging this current technical gap. In the process we examine the major historical contributions that have led to the formulation of Solomonoff Induction as well as criticisms of Solomonoff and induction in general. In particular we examine how Solomonoff induction addresses many issues that have plagued other inductive systems, such as the black ravens paradox and the confirmation problem, and compare this approach with other recent approaches. (shrink)

The technological singularity refers to a hypothetical scenario in which technological advances virtually explode. The most popular scenario is the creation of super-intelligent algorithms that recursively create ever higher intelligences. It took many decades for these ideas to spread from science fiction to popular science magazines and finally to attract the attention of serious philosophers. David Chalmers' (JCS 2010) article is the first comprehensive philosophical analysis of the singularity in a respected philosophy journal. The motivation of my article is to (...) augment Chalmers' and to discuss some issues not addressed by him, in particular what it could mean for intelligence to explode. In this course, I will (have to) provide a more careful treatment of what intelligence actually is, separate speed from intelligence explosion, compare what super-intelligent participants and classical human observers might experience and do, discuss immediate implications for the diversity and value of life, consider possible bounds on intelligence, and contemplate intelligences right at the singularity. (shrink)

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being (...) true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

Sequential decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameter-free theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline how the AIXI model can formally solve a number of problem classes, (...) including sequence prediction, strategic games, function minimization, reinforcement and supervised learning. The major drawback of the AIXI model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIXItl that is still effectively more intelligent than any other time t and length l bounded agent. The computation time of AIXItl is of the order t x 2^l. The discussion includes formal definitions of intelligence order relations, the horizon problem and relations of the AIXI theory to other AI approaches. (shrink)

The progression of theories suggested for our world, from ego- to geo- to helio-centric models to universe and multiverse theories and beyond, shows one tendency: The size of the described worlds increases, with humans being expelled from their center to ever more remote and random locations. If pushed too far, a potential theory of everything (TOE) is actually more a theories of nothing (TON). Indeed such theories have already been developed. I show that including observer localization into such theories is (...) necessary and sufficient to avoid this problem. I develop a quantitative recipe to identify TOEs and distinguish them from TONs and theories in-between. This precisely shows what the problem is with some recently suggested universal TOEs. (shrink)

Inspired by recent progress in multi-agent Reinforcement Learning (RL), in this work we examine the collective intelligent behaviour of theoretical universal agents by introducing a weighted mixture operation. Given a weighted set of agents, their weighted mixture is a new agent whose expected total reward in any environment is the corresponding weighted average of the original agents' expected total rewards in that environment. Thus, if RL agent intelligence is quantified in terms of performance across environments, the weighted mixture's intelligence is (...) the weighted average of the original agents' intelligences. This operation enables various interesting new theorems that shed light on the geometry of RL agent intelligence, namely: results about symmetries, convex agent-sets, and local extrema. We also show that any RL agent intelligence measure based on average performance across environments, subject to certain weak technical conditions, is identical (up to a constant factor) to performance within a single environment dependent on said intelligence measure. (shrink)

In this paper, we provide a detailed review of previous classifications of 2 × 2 games and suggest a mathematically simple way to classify the symmetric 2 × 2 games based on a decomposition of the payoff matrix into a cooperative and a zero-sum part. We argue that differences in the interaction between the parts is what makes games interesting in different ways. Our claim is supported by evolutionary computer experiments and findings in previous literature. In addition, we provide a (...) method for using a stereographic projection to create a compact 2-d representation of the game space. (shrink)

This paper develops a non-relativist version of contextualism about knowledge. It is argued that a plausible contextualism must take into account three features of our practice of attributing knowledge: (1) knowledge-attributions follow a default-and-challenge pattern; (2) there are preconditions for a belief's enjoying the status of being justified by default (e.g. being orthodox); and (3) for an error-possibility to be a serious challenge, there has to be positive evidence that the possibility might be realized in the given situation. It is (...) argued that standard "semantic" versions of contextualism (e.g. those of Lewis, Cohen, DeRose) fail to take these features into account, which makes them overly hospitable to the sceptic, and that Williams' version of contextualism, although incorporating (1), fails to do justice to (2) and (3). According to the contextualism developed here, although epistemic standards vary with the context, the truth-value of particular knowledge-attributions does not. Contexts here are understood as being constituted by two elements: an epistemic practice (a rule-governed social practice such as a scientific discipline, the law, a craft etc., in which knowledge-claims are evaluated according to specific standards) and the "facts of the matter" (i.e. those facts which, together with the epistemic standards in question, determine which error-possibilities are relevant and thus have to be eliminated for a knowledge-claim to be true). If there are several epistemic practices, and thus several contexts, in which a knowledge-claim can be evaluated, it is the "strictest" practice that counts. In this way, the counterintuitive consequence of other versions of contextualism that the same knowledge-claim can be true in one context, but false in another, can be avoided. At the same time, scepticism can be resisted since even in the "strictest" epistemic practices, error-possibilities become relevant only when backed by positive evidence that they might in fact obtain. (shrink)

Can you master the ideas of Plato, Kant, Nietzsche and Sartre? What does 'I think, therefore I am' really mean? Do you know the arguments for and against the existence of god? And what do the great philosophers tell us about knowledge and truth, good and evil? Packed with bite-sized briefings, shortcuts and bluffs, Get Smart: Philosophy demystifies 50 key philosophical concepts and provides you with all you need to speak out about the very biggest ideas.

Philosophy in Minutes distils 200 of the most important philosophical ideas into easily digestible, bite-sized sections. The core information for every topic - including debates such as the role of philosophy in science and religion, key thinkers from Aristotle to Marx, and introductions to morality and ethics - is explained in straightforward language, using illustrations to make the concepts easy to understand and remember. Whether you are perplexed by existentialism or pondering the notion of free will, this accessible small-format book (...) will help any reader to quickly grasp the basics of this highly nuanced subject. Chapters include: Truth and logic, Marxism, Communism and Socialism, Ontology, Philosophy and literature, Existence of God, Feminist theory, Consciousness, The future of philosophy. (shrink)

Quick, accessible, compact guide to understanding key political concepts. Contents include: Liberty, Justice, Equality, Human rights, Social contract, Democracy, Monarchy, Anarchism, Capitalism, Socialism, Nationalism and Globalisation.

Demystifies complex philosophical ideas and debates in a comprehensive format that introduces the major theories of such great philosophers as Socrates, Aristotle, Plato, Friedrich Nietzsche, Epicurus, and Thomas Aquinas.

ABSTRACTResearch that dissociates different types of processes within a given task using a processing tree approach suggests that attitudes may be acquired through evaluative conditioning in the absence of explicit encoding of CS-US pairings in memory. This research distinguishes explicit memory for the CS-US pairings from CS-liking acquired without encoding of CS-US pairs in explicit memory. It has been suggested that the latter effect may be due to an implicit misattribution process that is assumed to operate when US evocativeness is (...) low. In the present research, the latter assumption was supported neither by two high-powered experiments nor by complementary meta-analytic evidence, whereas evocativeness exerted an influence on explicit memory. This pattern of findings is inconsistent with the view that CS-liking acquired without encoding of CS-US pairs in explicit memory reflects an implicit misattribution process at learning. Hence, the underlying learning process is awaiting further empirical scrutiny. (shrink)

Researchers have long suspected that grapheme-color synaesthesia is useful, but research on its utility has so far focused primarily on episodic memory and perceptual discrimination. Here we ask whether it can be harnessed during rule-based Category learning. Participants learned through trial and error to classify grapheme pairs that were organized into categories on the basis of their associated synaesthetic colors. The performance of synaesthetes was similar to non-synaesthetes viewing graphemes that were physically colored in the same way. Specifically, synaesthetes learned (...) to categorize stimuli effectively, they were able to transfer this learning to novel stimuli, and they falsely recognized grapheme-pair foils, all like non-synaesthetes viewing colored graphemes. These findings demonstrate that synaesthesia can be exploited when learning the kind of material taught in many classroom settings. (shrink)

Das Verhaltnis von Spekulation und Vorstellung in Hegels enzyklopadischem System ist von einer eigentumlichen Spannung gekennzeichnet. Spekulation ist nicht nur die kritische Reinigung der Inhalte der Vorstellung von ihren Vorurteilen und Irrtumern, sondern zugleich Anerkennung des endlichen, vorstellenden Denkens innerhalb seiner Grenzen, sowohl auf dem Gebiet der Erkenntnistheorie als auch der Religion. Wie verhalten sich aber Spekulation und Vorstellung zueinander in verschiedenen Teilen des enzyklopadischen Systems, in der Logik und der Philosophie des Geistes? Diesem Zusammenhang gehen die Beitrage des Tagungsbandes (...) im Detail nach und uberprufen kritisch das enzyklopadische Zusammenspiel von Spekulation und Vorstellung in den Spharen des subjektiven und des absoluten Geistes. Mit Beitragen von: Kazimir Drilo, Markus Gabriel, Christoph Halbig, Gunnar Hindrichs, Axel Hutter, Franz Knappik, Anton Friedrich Koch, Christian Georg Martin, Jan Rohls, Rainer Schafer. (shrink)

The Conference on Artificial General Intelligence is the only major conference series devoted wholly and specifically to the creation of AI systems possessing general intelligence at the human level and ultimately beyond. Its second installation, AGI-09, in Arlington, Virginia, March 6-9, 2009, attracted 67 paper submissions, which is a substantial increase from the previous year. Of these submissions, 33 (i.e., 49%) were accepted as full papers for presentation at the conference. Additional 13 papers were included as position papers. The program (...) also included a keynote address by Juergen Schmidhuber on 'The New AI', a post-conference workshop on 'The Future of AI', and a number of pre-conference tutorials on various topics related to AGI. (shrink)

Kant’s revolutionary new approach to philosophy was accompanied by the introduction of a largely novel terminology. With the Kant-Lexikon, a lexical reference gives the modern reader access to his work on the basis of present-day editions and takes into account 20th century and contemporary research and advances in lexicology. The Kant-Lexikon includes 2395 entries authored by 221 scholars.

Marcus Giaquinto presents an investigation into the different kinds of visual thinking involved in mathematical thought, drawing on work in cognitive psychology, philosophy, and mathematics. He argues that mental images and physical diagrams are rarely just superfluous aids: they are often a means of discovery, understanding, and even proof.

In this article I have two primary goals. First, I present two recent views on the distinction between knowledge-that and knowledge-how (Stanley and Williamson, The Journal of Philosophy 98(8):411–444, 2001; Hetherington, Epistemology futures, 2006). I contend that neither of these provides conclusive arguments against the distinction. Second, I discuss studies from neuroscience and experimental psychology that relate to this distinction. Having examined these studies, I then defend a third view that explains certain relevant data from these studies by positing the (...) double dissociation of knowledge-that and knowledge-how and that is also able to do explanatory work elsewhere. (shrink)

I examine how co-parents should handle differing commitments about how to raise their child. Via thought experiment and the examination of our practices and affective reactions, I argue for a thesis about the locus of parental authority: that parental authority is invested in full in each individual parent, meaning that that the command of one parent is sufficient to bind the child to act in obedience. If this full-authority thesis is true, then for co-parents to command different things would be (...) for them to contest one another’s authority. The only course that respects the authority of both parents is for co-parents to agree to command the same thing. Further, what is commanded must not result from a ‘capitulation’ by one co-parent, rather, it should result from a compromise. Parental authority involves a duty to deliberate about which commands it is best to give the child. If a command results from a capitulation, one parent will rightly think of themselves as not having fulfilled their parental duty. Parental compromises are not best understood as bargains or conflicts, but by the metaphor of gifts given by each parent out of respect for the other’s authority. (shrink)

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking (...) in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking. (shrink)

The call for business practices that create benefits for companies, customers, and society is getting louder. This article analyzes a new implementation of such a win–win–win approach: the carrotmob. Activists and managers jointly organize a shopping flashmob in which consumers collectively purchase the products of a target company to reward its intent to act more socially responsible. Given that carrotmobs are only efficient if they are supported by a critical mass of consumers, a survey study of 337 young consumers explores (...) the critical drivers of carrotmob participation. Accordingly, object-oriented, personal, and social motives jointly determine carrotmob participation with social motives having the strongest impact. (shrink)

How homogenous are the sources of human motivation? Textbook Humeans hold that every human action is motivated by desire, thus any heterogeneity derives from differing objects of desire. Textbook Kantians hold that although some human actions are motivated by desire, others are motivated by reason. One question in this vicinity concerns whether there are states such that to be in one is at once take the world to be a certain way and to be motivated to act: the state-question. My (...) question here is different: whether passion and reason constitute distinct sources of human motivation: the source question. In this essay, I defend an affirmative answer to the source-question while remaining neutral on the state-question. I distinguish between what I call orectic desires, which are associated with the appetites, and anorectic desires, which are associated with judgments of the good. I argue that the two sorts of desires constitute distinct sources of motivation initially on the basis of their differing epistemological profiles. Specifically, self-attributions of anorectic desires are governed by the transparency condition, self-attributions of orectic desires are not. It emerges from this discussion that the motivation for performing an action arises in very different ways from each sort of desire. This difference, in turn, is explained by their differing relations to the will. Roughly, an orectic desire influences the will, but an anorectic desire is the will. The motivation to perform some of our actions derives simply from the determination that they are to-be-done—from reason. (shrink)

Many of Margaret Cavendish’s criticisms of Thomas Hobbes in the Philosophical Letters (1664) relate to the disorder and damage that she holds would result if Hobbesian pressure were the cause of visual perception. In this paper, I argue that her “two men” thought experiment in Letter IV is aimed at a different goal: to show the explanatory potency of her account. First, I connect Cavendish’s view of visual perception as “patterning” to the “two men” thought experiment in Letter IV. Second, (...) I provide a potential reply on Hobbes’s behalf that appeals to physiological differences between perceivers’ sense organs, drawing upon Hobbes’s optics in De homine. Third, I argue that such a reply would misunderstand Cavendish’s objective of showing the limited explanatory resources available in understanding visual perception as pressing when compared to her view of visual perception as patterning. (shrink)

In the Critique of Pure Reason, Kant famously criticizes traditional metaphysics and its proofs of immortality, free will and God's existence. What is often overlooked is that Kant also explains why rational beings must ask metaphysical questions about 'unconditioned' objects such as souls, uncaused causes or God, and why answers to these questions will appear rationally compelling to them. In this book, Marcus Willaschek reconstructs and defends Kant's account of the rational sources of metaphysics. After carefully explaining Kant's conceptions (...) of reason and metaphysics, he offers detailed interpretations of the relevant passages from the Critique of Pure Reason in which Kant explains why reason seeks 'the unconditioned'. Willaschek offers a novel interpretation of the Transcendental Dialectic, pointing up its 'positive' side, while at the same time it uncovers a highly original account of metaphysical thinking that will be relevant to contemporary philosophical debates. (shrink)

Following in a psychological and musicological tradition beginning with Leonard Meyer, and continuing through David Huron, we present a functional, cognitive account of the phenomenon of expectation in music, grounded in computational, probabilistic modeling. We summarize a range of evidence for this approach, from psychology, neuroscience, musicology, linguistics, and creativity studies, and argue that simulating expectation is an important part of understanding a broad range of human faculties, in music and beyond.

This book treats modal logic as a theory, with several subtheories, such as completeness theory, correspondence theory, duality theory and transfer theory and is intended as a course in modal logic for students who have had prior contact with modal logic and who wish to study it more deeply. It presupposes training in mathematical or logic. Very little specific knowledge is presupposed, most results which are needed are proved in this book.

A wide range of non-pharmaceutical interventions (NPIs) have been introduced to stop or slow down the COVID-19 pandemic. Examples include school closures, environmental cleaning and disinfection, mask mandates, restrictions on freedom of assembly and lockdowns. These NPIs depend on coercion for their effectiveness, either directly or indirectly. A widely held view is that coercive policies need to be publicly justified—justified to each citizen—to be legitimate. Standardly, this is thought to entail that there is a scientific consensus on the factual propositions (...) that are used to support the policies. In this paper, we argue that such a consensus has been lacking on the factual propositions justifying most NPIs. Consequently, they would on the standard view be illegitimate. This is regrettable since there are good reasons for granting the state the legitimate authority to enact NPIs under conditions of uncertainty. The upshot of our argument is that it is impossible to have both the standard interpretation of the permissibility of empirical claims in public justification and an effective pandemic response. We provide an alternative view that allows the state sufficient room for action while precluding the possibility of it acting without empirical support. (shrink)

Philosophy and Phenomenological Research, EarlyView.

Philosophy and Phenomenological Research, EarlyView.

This article argues that existing approaches to programming ethical AI fail to resolve a serious moral-semantic trilemma, generating interpretations of ethical requirements that are either too semantically strict, too semantically flexible, or overly unpredictable. This paper then illustrates the trilemma utilizing a recently proposed ‘general ethical dilemma analyzer,’ GenEth. Finally, it uses empirical evidence to argue that human beings resolve the semantic trilemma using general cognitive and motivational processes involving ‘mental time-travel,’ whereby we simulate different possible pasts and futures. I (...) demonstrate how mental time-travel psychology leads us to resolve the semantic trilemma through a six-step process of interpersonal negotiation and renegotiation, and then conclude by showing how comparative advantages in processing power would plausibly cause AI to use similar processes to solve the semantic trilemma more reliably than we do, leading AI to make better moral-semantic choices than humans do by our very own lights. (shrink)

ABSTRACTRecent research into evaluative conditioning shows that information about the relationship between the conditioned and unconditioned stimuli can exert strong effects on the size and di...

All the notes to the Farquharson translation, amplifying the twelve books of the "Meditations," are included in this volume.

Marcus Giaquinto tells the compelling story of one of the great intellectual adventures of the modern era: the attempt to find firm foundations for mathematics. From the late nineteenth century to the present day, this project has stimulated some of the most original and influential work in logic and philosophy.